moment based formulation of Navier-Maxwell slip boundary condition

Hello everyone,

I am reading a recent paper by T. Reis and P. J. dellar in Physics of Fluids( They have given the expression for the unknowns at the south boundary f2, f5 and f6 in equation 30(a), 30(b) and 30© respectively. Following the mathematical method explained by them i got the same expression for f2. But i think the authors have missed “-rho * uslip /2” term in equation 30(b) and “rho *uslip/2” term in equation 30©. It is very possible that i am completely wrong. But i think without these terms it will not be possible to satisfy tangential velocity
boundary condition.


Yes, you are right. There should be +/-rho*uslip/2 terms there. You spotted a typo that the authors missed in the proof-reading!!

Thanks a lot Pleb

I am also getting a miss-match in the slip velocity expression at the south boundary given in eqn 31. I am getting the bracketed term in the denominator as (1 + 2tau - 6sigma* l) .

The unknowns at the south-west corner are given in eqn 32(a) to 32(e). The expressions for f1, f2 and f8 are fine. I am getting the first term of RHS of eqn 32© as " - 2 *rho_in /3" and the bracketed part of last term as “u_slip - 1”. Also, in the expression for f6 (eqn 32d), the bracketed part of last term as “u_slip + 1”.

I will be glad if you can confirm it, waiting for your response.

32© should be -2rho/3, but I think the rest is ok. In your fist message I think f5 (30b) should have the +rhou/2 term, and f6 the negative.
For eqn 31, rhou=-3sigmaLPi/tau.
Since Pi=2taubar{Pi}/(2tau+1)
and bar{Pi}=rho
u-f1+f3+2f7-2f8 (where all f are fbar)
you get
where the … are the f terms. I think this rearranges into eqn 31.

Thanks a lot sir. Sorry sir, I did not know that you are the first author of the paper.

The problem in the “sign” of the terms was becuase of a horrible mistake( in the sign of components of microscopic velocity vector), which I was previously doing. Now i am getting eqn 31 as the expression for slip velocity. As you have mentioned everything is OK apart from
“rho * u_slip/2” in eqn 30 (b), “-rho * u_slip/2” in eqn 30 © and “-2 * rho/3” in eqn 32©.

Sorry for the trouble sir and thanks a lot for the clarification.

No problem at all - I apologise for the typo!
Good luck,